if it is held about as shown at _B_, when turning the flange
casting _C_, the surfaces _s_ and _s_{1}_ can be finished without
changing the tool's position. Cylindrical and radial surfaces are often
turned in this way in order to avoid shifting the tool, especially when
machining parts in quantity.

=Tool Grinding.=--In the grinding of lathe tools there are three things
of importance to be considered: First, the cutting edge of the tool (as
viewed from the top) needs to be given a certain shape; second, there
must be a sufficient amount of clearance for the cutting edge; and
third, tools, with certain exceptions, are ground with a backward slope
or a side slope, or with a combination of these two slopes on that part
against which the chip bears when the tool is in use.

[Illustration: Fig. 9. (A) The Way in which Tool is sometimes displaced
by Thrust of Cut, when set at an Angle. (B) Tool Set for Finishing both
Cylindrical and Radial Surfaces]

In Fig. 10 a few of the different types of tools which are used in
connection with lathe work are shown. This illustration also indicates
the meaning of the various terms used in tool grinding. As shown, the
clearance of the tool is represented by the angle [alpha], the back
slope is represented by the angle [beta], and the side slope by the
angle [gamma]. The angle [delta] for a tool without side slope is known
as the lip angle or the angle of keenness. When, however, the tool has
both back and side slopes, this lip angle would more properly be the
angle between the flank _f_ and the top of the tool, measured diagonally
along a line _z--z_. It will be seen that the lines _A--B_ and _A--C_
from which the angles of clearance and back slope are measured are
parallel with the top and sides of the tool shank, respectively. For
lathe tools, however, these lines are not necessarily located in this
way when the tool is in use, as the height of the tool point with
relation to the work center determines the position of these lines, so
that the _effective_ angles of back slope, clearance and keenness are
changed as the tool point is lowered or raised. The way the position of
the tool affects these angles will be explained later.

[Illustration: Fig. 10. Illustration showing the Meaning of Terms used
in Tool Grinding as applied to Tools of Different Types]

While tools must, of necessity, be varied considerably in shape to adapt
them to various purposes, there are certain underlying pr